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I have an ordinal, categorical variable with five levels, of which the last two have only one observation for each. Should I leave them alone, omit them, incorporate them in another category, or do some other thing?

More generally, what is the strategy to address the situation where responses are too heavily skewed? I'd like as well to know reference readings if you have some.

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Where there's the smoke of an ordinal variable, there's the fire of a latent continuous variable smoldering beneath it. If you can conceive of such a latent variable in this case, NonSleeper, then you have the opportunity to view your problem as one of interval censoring. To make this idea concrete, imagine you did a clinical trial on a very tight budget, where you could only afford a broken digital scale to weigh the subjects. The LCD display is damaged, such that only the 100's-place digit can be read. Your subjects' weights thus were coded 0 (<100lbs), 1 (100-199lbs), 2 (200-299lbs) and 3 (300+lbs). I note that CRAN lists several R packages that allow you to estimate certain types of model in the presence of interval censoring, although my own approach to interval censoring has been simply to use Bayesian methods that allow me flexibly to express my prior knowledge about the latent variable (say, a known, age-dependent distribution of weight in the population from which study subjects were drawn), and to incorporate other study measures (say, waist circumference) in a theoretically grounded way, to achieve the most efficient use of all study data.

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Whether to incorporate them depends on your question of interest and whether you can re-formulate it to include that '3+' category, whether it would be more or less informative than omitting these levels. Too few observations in a category is not commonly referred to as skewness, so the second question is not a generalization of the first. The answer will depend on the type/degree of the skewness. For example, rank-based methods will often work for skewed data; transformations can mitigate skewness. There are also approaches for zero-rich data. Here is one example: https://www.maths.otago.ac.nz/home/resources/david_fletcher/Fletcher_et_al_2005.pdf

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If your variable is highly skewed try applying a Box Cox transformation to normalize the skewness http://stat.ethz.ch/R-manual/R-patched/library/MASS/html/boxcox.html

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